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Signed integers  with ordinary addition and multiplication form the prime example of a  ring.  The original motivation for  Ring Theory  was to investigate what properties of the integers are retained in other sets similarly endowed with an additive operator and a multiplication distributive over it.  (Only for integers can multiplication be defined as repeated addition.)
 
The locution  distributive ring  merely distinguishes the  algebraic term  studied here from other meanings of the word  ring  itself.  There's no such thing as a  "non-distributive ring". The term  associative ring  is also used  (non-associative rings are better called  algebras).

(2019-04-20)   A Brief History of Rings
(Assuming some familiarity with the  modern concept of abstract rings.)

Abstract  rings  were introduced in 1871 by  Richard Dedekind (1831-1916)  under the name of  Ordnung.  Dedekind also pioneered the related notions of  module  and  ideal  (then limited to  prime ideals)  in the context of what we would now recognize as the integers of a number field.

In 1892,  David Hilbert (1862-1942)  coined the modern name  (Zahlring,  then   Ring)  which first appeared in print in 1897.  The German word  Ring  and the English word  ring  share with  circle  two connotations which may have inspired Hilbert:  One denotes a group of people,  the other evokes circularity and looping back  (as in the rings of nonzero characteristics  which Hilbert was then concerned with).  The latter etymology is dominant,  as shown by the fact that it's solely responsible for the French translation  (anneau)  which lacks the first connotation.

In 1914,  Abraham Fraenkel (1891-1965)  introduced an axiomatic definition which was far more restrictive than the  current one,  as it required the existence of a multiplicative identity and postulated that every noninvertible element should be a  divisor of zero  (which rules out  ordinary integers).

In 1917,  the Japanese mathematician Masazo Sono (1886-1969)  dropped those requirements but retained commutativity,  quoting compatibility with the  field axioms  published in 1903 by  L. Eugene Dickson (1874-1954).

In 1921,  Emmy Noether (1882-1935)  did the same thing independently.  A few years later,  she finally dropped the requirement of commutativity.

Rings with a multiplicative neutral element are the most important type of rings and  many authors  will only study those.  They're variously known as  "rings with 1",  "rings with unity",  "rings with identity",  "unit rings",  "unitary rings"  or  "unital rings"  (which is the only term I'll use).

Do forsake the linguistic monstrosity  "rng"  ("ring without i", pronounced "rung").  This joke was launched in 2005 as a Wikipedia stub which has now morphed into a full hoax,  mascarading as  fait accompli,  claiming adamantly that the word  ring  should only be used for unital rings.
 
The neurotic credo of the original perpetrator  (who signed KSmrq)  was addressed  a few months later  to  Bernard Haisch:
 
" You do not get to choose whether or not an article on you appears in Wikipedia, and you have no veto power over its contents.  The article can cast you as a genius or an imbecile, a respected scientist or a crackpot...  a vandal could replace a page, any page, with total gibberish.  The page on Einstein might have a statement inserted to the effect that he was a Nazi collaborator, or that his theories have been totally discredited, or that he was a silicon-based life form from Proxima Centauri...  Wikipedia does not operate by your rules but by its own conventions;  I suggest you learn to accept it...  I can assure you resistance is futile. "  [sic]
 
Well,  whoever shows so little respect for the feelings of fellow human beings can't be expected to care much for the integrity of mathematical discourse...  Mercifully,  nobody takes seriously a similar joke stating that a  semiring  ought to be called a  "rig"  (ring without negatives).

For any given  (abelian)  additive group  G,  the  ring of square zero  G(0)  is the ring in which the product of two elements is always zero.  Bourbaki  had to call it a  pseudo-ring of square zero   (pseudo-anneau de carré nul)  because,  in their work,  the bare term  ring  was reserved for  unital rings.  This trivial case plays a key role in the  enumeration  of finite rings.

Reputable authors will routinely start a specialized discussion with a fair warning  like:  "Throughout this paper,  the symbol  R  will denote a finite associative ring with multiplicative identity."  That's the way to go...

(2006-02-15)   Rings   (called  distributive rings  for disambiguation).
Addition, subtraction and multiplication are defined,  division needn't be.

(A, +, . )  is a  ring  when  addition  (+)  and  multiplication  (.)  are  well-defined  internal operations over the set  A  with the following properties:

• (A,+)  is a commutative group  whose neutral element is called  0  (zero).
• Multiplication is an  associative  (not necessarily commutative)  internal operation which is  distributive  over addition.  That's to say:
  "x   "y   "z       x.(y + z)   =   x.y + x.z       and       (x + y).z   =   x.z + y.z

Multiplicative  notations allow the omission of the  dot  symbol  (.).

Optional properties of a ring can be indicated by specific qualifiers:

• Unital Ring :   There's a multiplicative neutral element:  1.x = x.1 = x
• Commutative Ring  :   "x   "y   x.y = y.x
• Integral Ring :   The product of two nonzero elements is nonzero.
• Division Ring :   Any nonzero element has a multiplicative inverse.

There seems to be universal agreement to define an integral domain  as a  commutative integral ring.  The current trend in  Ring Theory  (echoed by  Wikipedia)  is to call a  domain  any  integral ring  and specify that the term  integral domain  only applies to commutative ones.  I still advise against the term  domain  outside idioms like  integral domain.

This has the added benefit of compatibility with  topology,  where the bare term  domain  normally denotes a  connected open set.

A  field  is normally defined as a  commutative  division ring  (a division ring where multiplication is commutative)  unless otherwise specified.  I regard as synonymous the locutions  noncommutative division ring  and  skew field  (as well as the semi-acceptable  oxymoron  of  noncommutative field).  Some authors  allow  commutativity in a  skew field,  in part to translate what the French call a field  (corps)  which is a division ring,  commutative or not.

Lesser Rings :

For the record,  some algebraic structures have been defined which are endowed with an addition and a multiplication  distributive  over it,  but without some of the other requirements imposed on  rings.  Examples:

A  semiring  is built on an additive  monoid  instead of an additive  group.  This is to say that a  semiring  contains a zero element  (neutral for addition)  but subtraction need not be well-defined.  In a semiring,  0  is  postulated  to be  absorbent  ("x,   0.x = x.0 = 0)  which is a  theorem  in a ring.  The prototypical example of a semiring is   (,+,×)   the set of  natural integers,  endowed with usual addition and multiplication.

  =   { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... }

Also common are the  tropical semirings  (e.g., the  max-plus algebra  where  -¥  is included as neutral additive element with the new operations defined by  xÅy = max(x,y)   and   xÄy = x+y).  Likewise in  idempotent semirings  xÅx = x  holds for any  x.  Those may be called  semirings of characteristic one  (the only  ring  of  characteristic  one is the  trivial field).

A  near-ring  is a ring-like structure where the additive group is not necessarily  abelian  (i.e., addition needn't be commutative).  Multiplication must still be associative but distributivity may not hold on both sides.  Thus,  in a  right near-ring  multiplication is only required to  right-distribute  over addition,  which is to say:

"x   "y   "z       (x + y).z   =   x.z + y.z

For example,  over an  additive group  G  (abelian or not)  the  functions  from  G  to  G  form a  right near-ring  if the addition of functions is defined  pointwise  and multiplication is understood as the  composition  of functions:

"xÎG         ( f + g ) (x)   =   f (x)  +  g (x)             ( f . g ) (x)   =   f ( g (x) )

Those definitions imply  right-distributivity   ( f + g ) . h  =  f.h + g.h   since:

"xÎG         ( f + g ) (h (x))   =   f (h (x))  +  g (h (x))    

Ring   |   Semiring   |   Near-ring & Near-field   |   Near-semiring
 
Ring Examples (7:17)  by  Liliana de Castro  (Socratica, 2017-07-09).

(2006-02-15)   Divisors,  divisors of zero  and  proper  zero-divisors :
In some rings,  the product of two  nonzero  elements can be zero.

In a ring, by definition,  an element  d  is said to be a  divisor  of a given element  a  when there is a  nonzero  element  x  such that:

d x   =   a       or       x d   =   a

In particular  (with  a = 0 )  a  divisor of zero  is an element whose product into some nonzero element is equal to zero.  As usual,  a  proper divisor  can't be equal to the  dividend  itself.  So,  the zero element itself isn't a  proper  divisor of zero.  Zero is sometimes called a  trivial  divisor of zero.

The above definition implies that zero isn't at all a divisor of zero in the trivial ring  ({0},+,x)  because of the lack of nonzero elements in it.

• d  is a  left divisor of zero  when there's a nonzero  v  such that  dv = 0.
• d  is a  right divisor of zero  when there's a nonzero  u  such that  ud = 0.
• d  is a  two-sided divisor of zero  when both exist  (whether  u = v  or not).
• d  is a  regular element  if  neither  proposition is true.

Many authors,  including myself,  use the locution  zero-divisor  to denote a  proper divisor of zero  (more awkwardly dubbed  nontrivial divisor of zero  or  nonzero divisor of zero).  I strongly recommend the hyphenation to best indicate that this locution must be taken as a whole.

Likewise,  left zero-divisors or right zero-divisors are always understood to be nonzero...  There are no zero-divisors in  integral rings  (including  integral domains,  division rings,  fields  and  skew fields).

This convention is often implicitly made  (otherwise it wouldn't make much sense to speak of "rings without zero-divisors")  but it's prudent to state explicitly before a formal argument that a  zero-divisor  can't be zero  (even in the ring  {0}).  Conversely,  the locution  "divisor of zero"  can  be read grammatically as above,  which makes it include zero.

Two elements  x  and  y  are said to be  (mutually)  orthogonal  when:

x y   =   0         and         y x   =   0

If some  n^th  power of an element is zero, that element is said to be  nilpotent.  Clearly, a nonzero nilpotent element is a zero-divisor.

x ⁿ   =   0

The simplest example  is the residue  2  in the ring  / 4  (i.e.,  the ring formed by the four possible residues of integers modulo 4)  where:

2 ²   =   0

One example of a ring with  zero-divisors  but no  nilpotent  elements is the ring of  polyadic integers  for any radix  g  that's  not  a prime-power.

Regular element = cancellable element

Those are the names given to elements which are  not  divisisor of zero to a specified side.  Equivalently, muliplication to one side by an element which is regular to that side is  injective.  For example,  if the element  r  is  right-regular,  matching right-factors of  r  can be cancelled from an equation:

x r   =   y r       iff       x   =   y

No side is specified when a regular element is regular to both sides.

Divisor   |   Zero-divisor

(2019-05-11)   Units are  invertible  elements in a unital ring.

In a  finite  unital  ring,  every non-invertible element is a  divisor of zero.  To put it bluntly,  every element divides either zero or unity.

In 1914,  Abraham Fraenkel (1891-1965)  had imposed this property as an axiom for all rings,  which ruled out the  star of the show  (the ordinary integers)  since most integers are neither units nor divisors of zero.  The postulate is obsolete but it's a theorem for  finite  rings.

Proof :   For any element  a  of the  finite unital ring  A ¹ {0},  if the  map  which sends  x  to  a x  is injective,  then it's also surjective  (since  A  is  finite)  so there's an element  v  whose image is  1,  meaning  a v = 1.

On the other hand,  if that map isn't injective,  then two  different  elements  x  and  y  have the same image and  x-y = v  verifies  a v = 0  with  v ¹ 0.

Likewise,  considering the map which sends  x  to  x a,  we see that there's a nonzero element  u  such that  u a  is either  0  or  1.

We can't have  u a = 1  and  a v = 0  with a nonzero  v  (or else we'd have  v = u a v = 0)  or vice-versa.  Therefore,  either  a  is a unit  (with the same inverse on both sides  u = u a v = v)  or it's a two-sided divisor of zero.  

The units of a  unital ring  A  form a multiplicative groupe denoted  A*  or  U(A)  (sometimes  E(A)  as the German name for  unit  is  Einheit).  That's just a special case of the group  M*  formed by all the invertible elements in a multiplicative monoid  M,  so the  same notation  can be used.

Units   |   Associated elements   |   Divisibility
 
Units in a Ring (7:13)  by  Liliana de Castro  (Socratica, 2017-09-06).

(2006-02-15)   Ideal  I  in a Ring  A
An ideal is a  multiplicatively absorbent  additive subgroup.

An  ideal  is an additive subgroup that contains a product whenever it contains a  factor.  (Such a thing is called  multiplicatively absorbent,  or  absorbent  for short.)

A  subring  is a ring contained in another one  (using the same operations).  With the traditional definition of a ring adopted here  (where we don't require a ring to be unital)  a subring is simply a nonempty subset closed under  subtraction  and  multiplication.  That much is clearly true for an ideal.  Thus,  ideals can also be defined as  absorbent subrings.

However,  a subring (or an ideal) of a  unital  ring need not be unital itself.  Therefore,  those who are adamant that only unital rings should be called rings will reject those subring characterizations.  Their loss.

For a  left-ideal  I, the product  ax  is in I whenever x is:   "aÎA, aI Í I
For a  right-ideal  I, the product  xa  is in I whenever x is:   "aÎA, Ia Í I
Unless otherwise specified, an  ideal  is  both  a right-ideal and a left-ideal.

Every ring is a (two-sided) ideal of itself,  called the  unit ideal.  Any other ideal  (one-sided or two-sided)  is said to be  proper.

All rings with more than one element have at least one  proper ideal,  namely the single-element subring  {0}  which is called the  zero ideal.  The ring  {0}  doesn't have any proper ideals.  The  unit ideal  and the  zero ideal  are both called  trivial  (whether or not they are the same).

The sum and the intersection of two same-sided ideals are ideals on that side.

The  universal convention  due to  Hermann Minkowski (1864-1909)  is that the sum  (resp. product)  of two sets is defined to be the set whose elements are sums  (resp. products)  of elements from those two sets.

If  I  is a left-ideal and  J  is a right-ideal,  then  I.J  is a double-sided ideal,  but neither  J.I  nor  I Ç J  are necessarily one-sided ideals.

A  one-sided or two-sided ideal is called  maximal  when it's a  proper  ideal not contained in any other  proper  ideal of the same kind.  Thus,  the whole ring is never a  maximal ideal  of itself.  {0}  is maximal only if the ring is nonzero and doesn't have any other proper ideals of the relevant kind.

A ring  A  is said to be a  filial ring  when any ideal of an ideal of  A  is also an ideal of  A  itself.  Using  Wielandt's symbols  that's to say:

J ⊲ I ⊲ A   Þ   J ⊲ A

So,  ring  A  is  filial  when all its  2-accessible  subrings are ideals,  defining a subring  S  as  n-accessible  when a sequence  A_i  exists which verifies the following  (such an  S  is  precisely  n-accessible if not k-accessible for k<n):

S   =   A₀  ⊲  A₁  ⊲   ...   ⊲  A_n   =   A

The ideal  generated  (to one side)  by some set  X  is well-defined as the intersection of all ideals  (to that side)  containing  X,  since any intersection of ideals of a given kind is an ideal of the same kind.

An ideal  generated  by a single element is called a  principal ideal.  One example of such a right-ideal is the set  a A  of all right-multiples of the element  a  in the ring A  (e.g.,  2   is the ideal of all even integers).

A ring,  like ,  whose ideals are  all  principal is called a  principal ring.  Such a ring is called a  principal integral domain  (abbreviated PID)  if it has no proper divisors of zero  (i.e., the product of two nonzero elements is never zero).  Note that  Bourbaki  requires a  principal ring  to be a PID.

In a  field,  there are only two ideals,  namely  {0}  and the whole field.  They are both principal  (respectively generated by the elements  0  and  1)  so a field is a  PID.  (HINT:  If an ideal of a field contains a nonzero element it also contains  the product of that element by its inverse,  which is  1.)  Every  skew field  is a PID too.

Ideals were introduced in 1871 by  Richard Dedekind (1831-1916)  as he investigated what are now called  completely prime ideals,  namely ideals which don't contain a product  unless  they contain at least one  factor  (e.g.,  among integers, the multiples of a prime number  have that property).  In commutative rings,  those are just  prime ideals,  namely ideals which don't contain a product of two ideals unless they contain at least one of them.  A  completely prime ideal  is always a  prime ideal,  but the converse may not be true in the noncommutative case.  That distinction was introduced in 1928 by  Wolfgang Krull (1899-1971).

The radical  Rad(I)  of an ideal  I  is the set of all ring elements which have one of their powers in  I.  The radical of an ideal is an ideal.  If an ideal is the radical of another it's called a  radical ideal.  Every  prime ideal  is a  radical ideal.  Modulo a  radical ideal,  there are no  nilpotent  residues.

In particular,  the ideal  Rad({0}),  called the  nilradical  of  A,  is the set of all nilpotent elements of  A.  It's the intersection of all  prime ideals  of  A.

The  Jacobson radical  J(A)  of a ring  A  is the intersection of all the  maximal ideals  of  A.  Since all  maximal ideals  are  prime,  the  nilradical  is contained in the  Jacobson radical.

Ideal  |  Reduced ring  |  Nil ideal  |  Nilpotent ideal  |  Radical of a ring
Nilradical of a commutative ring  |  Jacobson radical (1945)  |  Nathan Jacobson (1910-1999)
 
Ideals in Ring Theory (11:56)  by  Liliana De Castro  (Socratica, 2020-02-18).

(2006-02-15)   Residue Ring  (modulo a given ideal I of a ring A)
The ring  A / I  which consists of all residue classes  modulo  I.

Modulo an ideal I of a ring A, the residue-class (or just  residue ) [x] of an element  x  of  A  is the set of all elements  y  of  A  for which  x-y  is in  I.

The set of all residues modulo I is denoted A/I.  It's a ring, variously called quotient ring, factor ring, residue-class ring or simply residue ring.

For example,  / 4  is the ring formed by the four residue classes modulo 4, whose addition and multiplication tables are shown at right.  (Note that "2" is a  nilpotent  divisor of zero.)

+ 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2

´ 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1

The ring   / p  =  ( / p, +, ×)   is a  field  if and only if  p  is  prime.

The notation  _p  instead of  / p  is  not  acceptable,  as the former is reserved for the  (infinite)  ring of  p-adic integers.

In particular,  the  Boolean field   / 2   has just two elements;  0  and  1  (called  bits  nowadays).  It's used  below  to construct a nontrivial ring,  which provides concrete examples of many abstract concepts.

(2019-04-19)   Ring Homomorphisms and Isomorphisms
Function  f  from one ring to another,  which respects the ring operators:

f (x+y)   =   f (x)  +  f (y)             f (x y)   =   f (x)   f (y)

If such an  homomorphism  is  bijective,  it's called an  isomorphism.  An isomorphism from one ring to itself is called an  automorphism.

Anti-homomorphisms  reverse multiplication:   f (x y)   =   f (y)   f (x)

If  f  is an homomorphism from ring  A  to ring  B  and  J  is a  subring  of  B,  then  I = f ^-1 (J)  is a subring.  It's an  ideal  of  A  if  J  is an ideal of  B.  I  is called the  contraction  of  J.

Kernel :   In particular,  f ^-1 ({0_B})  is an ideal,  called the  kernel  of  f :

ker  f     =     f ^-1 ({0})     =     { xÎA  |  f (x) = 0 }

Kernel of a group homomorphism   |   Categorial kernel

(2006-06-13)   Characteristic of a Ring  A :   p = char(A).
It's either  0  or the least  p>0  for which all sums of  p  like terms vanish.

In a  unital  ring  A,  we may call "1" the neutral element for multiplication and  name  the elements of the following sequence after ordinary  integers:

1 º 1,   2 º 1+1,   3 º 1+1+1,   4 º 1+1+1+1,   5 º 1+1+1+1+1,   ...

If all the elements in this sequence are nonzero, the ring is said to have zero characteristic.  Otherwise, the vanishing  indices  are multiples of the least of them,  which is called the  characteristic  of the ring, denoted  char(A).

The only ring of characteristic  1  is the  trivial field  (where  1 = 0).

The characteristic of a nontrivial unital ring without zero-divisors is either zero or a  prime number.  (HINT:  any  integer  (1+1+...)  corresponding to a  prime divisor  of a  composite  characteristic would be a  zero-divisor.)

In particular, the characteristic of any nontrivial field  (or skew-field)  is either  0  or a prime number.

The  characteristic  of a non-unital ring is defined as the least positive integer  p  such that a sum of  p  identical terms always vanishes  (if there's no such  p,  then the ring is said to have zero characteristic).

The characteristic of a ring depends  only  on its additive group  (e.g.,  if that additive group is the  cyclic group  C_n ,  then the characteristic is  l(n),  the  reduced totient  of  n).

A  finite  ring can't have zero characteristic.  The characteristic of a finite ring always divides its number of elements.  (cf. structure of abelian groups).

Frobenius Map  (1880):

If the characteristic  p  of a  commutative ring  is a  prime number,  we have:

( x y ) ^p   =   x ^p y ^p         and         (x + y) ^p   =   x^p + y^p

The former relation is due to  commutativity.  The latter relation comes from  Newton's binomial formula,  with the added remark that the binomial coefficient  C(p,k)  is divisible by  p,  if  p  is prime,  unless  k  is  0  or  p.

Thus,  the map defined by   F(x)  =  x^p   respects  both addition and multiplication.  This  ring automorphism,  is called the  Frobenius map,  in honor of  F. Georg Frobenius  (1849-1917)  who discovered the relevance of such things to  algebraic number theory,  in 1880.

The automorphism group of the Galois field  GF(pⁿ )  is a  cyclic group  of order  n,  generated  by the above  Frobenius map.

(2019-01-27)   The SUN ring   (Smallest Unital Noncommutative ring).
It can be represented by  8  triangular  2 by 2  boolean  matrices.

We may construct it as a  vector space  over the integers  modulo 2  by choosing any basis of three independent matrices.  This gives an additive group  (of  characteristic  2)  isomorphic to the  8  integers from  0  to  7  endowed with  bitwise  addition  (just like the additive group of the  Galois field  GF(8)  which see for the explicit  addition table).

We'll use the  ring isomorphism  defined by the following correspondence between powers of two and lower-triangular binary 2×2 matrices  (we also give equivalent upper-triangular matrices  highlighted in yellow  to make explicit the isomorphism between the two types of triangular matrices). 

Upper Triangular	1  0  0  1	0  1  0  0	1  0  0  0
Lower Triangular	1  0  0  1	0  0  1  0	0  0  0  1
Binary Name	1	2	4

Choices for  1  and  2  were engineered to give the names  0,1,2 and 3  to the four elements whose unique properties make them invariant under  any  ring automorphism.  Those form a  commutative subring  whose  multiplication table  is thus made identical to the multiplication table of  / 4.

Four possible matrices  can then be assigned to  4.  I picked one arbitrarily,  as I'm still looking for a good reason to distinguish one as  canonical.

×			0  0  0  0	1  0  0  1	0  1  0  0	1  1  0  1	1  0  0  0	0  0  0  1	1  1  0  0	0  1  0  1
			0  0  0  0	1  0  0  1	0  0  1  0	1  0  1  1	0  0  0  1	1  0  0  0	0  0  1  1	1  0  1  0
			0	1	2	3	4	5	6	7
0  0  0  0	0  0  0  0	0	0	0	0	0	0	0	0	0
1  0  0  1	1  0  0  1	1	0	1	2	3	4	5	6	7
0  1  0  0	0  0  1  0	2	0	2	0	2	0	2	0	2
1  1  0  1	1  0  1  1	3	0	3	2	1	4	7	6	5
1  0  0  0	0  0  0  1	4	0	4	2	6	4	0	6	2
0  0  0  1	1  0  0  0	5	0	5	0	5	0	5	0	5
1  1  0  0	0  0  1  1	6	0	6	2	4	4	2	6	0
0  1  0  1	1  0  1  0	7	0	7	0	7	0	7	0	7

Let's use this example as an opportunity to review the basic concepts:

• Element  1  is neutral for multiplication:  1 x  =  x 1  =  x   for any x.
• Element  2  is  nilpotent.  {0,2}  is a  ring of square zero.
• As  expected,  all elements are either  divisors of zero  or  units  (1 and 3).
• Besides  2  and  3,  all elements are  idempotent.
• 4  and  5  are  mutually orthogonal.  So are  6  and  7.
• The two  trivial  ideals  are  {0}  and the  unit ideal  {0,1,2,3,4,5,6,7}.
• The  zero ideal  {0}  is the only  proper  trivial ideal  (as always).
• The only  nontrivial  two-sided  ideals are  {0,2},  {0,2,4,6}  and  {0,2,5,7}.
• Besides the two-sided ideals,  the only left-ideals are  {0,4}  and  {0,6}.
• Besides the two-sided ideals,  the only right-ideals are  {0,5}  and  {0,7}.
• The only  maximal  ideals are  {0,2,4,6}  and  {0,2,5,7}.
• The  subrings  which are not single-sided ideals are  {0,1}  and  {0,1,2,3}.
• {0,2,4,5,6,7}  is multiplicatively absorbent but it isn't an additive subgroup.
• There are  2  ring automorphisms  and  2  anti-automorphisms,  namely:

The automorphisms and anti-automorphisms of the  SUN  ring :

	0	1	2	3	4	5	6	7
Identity	0	1	2	3	4	5	6	7
Automorphism	0	1	2	3	6	7	4	5
Anti-automorphisms	0	1	2	3	5	4	7	6
	0	1	2	3	7	6	5	4

As both anti-automorphisms are involutions,  they can be used to give  SUN  the structure of an  involutive ring,  in two different ways.  Using the  Cayley-Dickson construction  with either choice yields an algebraic stucture of 16 elements which is  not  a ring  (the composite structure isn't multiplicatively associative since basic multiplication isn't commutative).

If  f (x)  is the number of  1's  in the diagonal  of  [the matrix for]  x,  then:

f (x y)   ≤   min ( f (x),  f (y) )

´	1	i	j
1	1	i	j
i	i	0	0
j	j	j	j

More generally,  for any  commutative  unital ring  A,  we may use the above pattern to define  SUN(A)  as a  3-dimensional  module  over  A,  where multiplication is determined by the multiplication table at right,  for a basis  (1,i,j)  of the module.

For short,  SUN_q = SUN( GF(q) )  when  q  is a prime-power.
Furthermore,  SUN = SUN₂  denotes the ring discussed above.

For a prime  p,  SUN_p  is the only unital noncommutative ring of order  p³.

---

Alternately,  we could have chosen to define the  SUN  ring as generated by two  distinct  elements  u  and  v  obeying the following relations:

u²  =  u         v²  =  v         u v  =  v         v u  =  u

With the above numbering,  that's only true when  {u,v} = {4,6}.

We could also have used the relations below,  characterizing  {u,v} = {5,7} :

u²  =  u         v²  =  v         u v  =  u         v u  =  v

Smallest non-commutative ring with unity  Mathematics Stack Exchange  (2013-12-25).
Smallest order for a non-commutative ring with unity  by  Andrew Qian  (Quora, 2015-01-29).

(2019-02-14)   Idempotent elements  &  Peirce decompositions   (1870)
An  idempotent  element splits a ring into a direct sum of  4  components.

An  idempotent  element of a ring  A  is a solution of the equation:

x ²   =   x

A  Boolean ring  is a ring where  every  element is idempotent.  A  unital  Boolean ring is called a  Boolean algebra.

Every ring has at least one idempotent element  (namely  0)  and every  unital ring  with more than one element has another trivial one  (namely  1).

If a unital ring has other idempotent elements  (said to be  nontrivial)  then it has at least one  zero-divisor  because,  in a unital ring,  the above equation reduces to the following zero product of two factors,  neither of which is zero when  x  is neither 0 nor 1:

x (1-x)   =   0

Left-multiplication or right-multiplication by an  idempotent  is analogous to a geometrical  projection  (projecting a projected image doesn't change it).  This geometrical analogy suggests that a ring can be expressed as a direct sum of the images of two such complementary projections,  as shown next.

Peirce Decompositions  (Benjamin Peirce, 1870) :

If  e  is an  idempotent  of the unital ring  A  (i.e.,  e² = e)  then  A  can be split into a  direct sum  in two different ways  (using Minkowski notations):

• The  left Peirce decomposition :   A   =   e A  Å  (1-e) A
• The  right Peirce decomposition :   A   =   A e  Å  A (1-e)

In the noncommutative case,  those two are distinct and we may apply them successively to obtain the full  (two-sided)  Peirce decomposition  of  A  into four components,  best presented formally as a  matrix:

A   @
			e A e (1-e) A e		e A (1-e) (1-e) A (1-e)

The right-hand-side denotes a set of  2×2  matrices whose coefficients are in four subsets of  A  specified by Minkowski operations.  Addition and multiplication of such matrices reduce to component operations in  A  according to the  usual rules.  (It's a simple exercise to show that this set of matrices is stable under addition and multiplication.)

The decomposition can be  trivial  (e.g.,  with  e = 1).  The reader may want to work out the example   e = 4  with numbered elements of the  SUN ring,  which recovers a representation in terms of triangular boolean matrices:

SUN2   =   A   @

4 A 4 5 A 4

4 A 5 5 A 5

=

{0,4} {0}

{0,2} {0,5}

More generally,  what we did for   e₁ = e   and   e₂ = 1-e   can be done for any set of  n  pairwise  orthogonal  idempotents  adding up to unity:

e_i  e_j   =  d_ij e_j               e₁  +  e₂  +  ... +  e_n   =   1

A   @
			e1 A e1 e2 A e1 ... en A e1		e1 A e2 e2 A e2 ... en A e2		... ...   ...		e1 A en e2 A en ... en A en

A more intricate decomposition for  Jordan algebras  was introduced in 1947 by  A.A. Albert (1905-1972).

Peirce decompositions, idempotents and rings  by  P.N. Ánk,  G.F. Birkenmeier,  L. Van Wyk  (2017-02-20).
 
Idempotents   |   Peirce decomposition   |   Peirce decomposition in a non-unital ring

The terms  idempotent  and  nilpotent  were coined in 1870 by the American mathematician Benjamin Peirce (1809-1880)  who bore the- same name as his father  Benjamin Peirce (1778-1831)  head of the  Harvard library  for the last five years of his life.
 
The younger  Benjamin Peirce  (rhyming with  terse  or  purse)  taught mathematics at  Harvard University  for nearly 50 years.  In 1832,  he proved that an odd perfect number  (if such a thing exists)  can't have fewer than  4  distinct prime factors  (that's been upgraded to 10, now).
 
So far,  the  Mathematics Genealogy Project,  found 6038 doctoral descendants of his.  This writer descends from him through a lineage of seven academic generations,  namely
 
Benjamin Peirce (1809-1880).  A.B. Harvard 1829, under Nathaniel Bowditch (1773-1838).
Joseph Lovering (1813-1892).  A.B. Harvard 1833.
John Trowbridge (1843-1923).  S.D. Harvard 1873.
George Washington Pierce (1872-1956).  Ph.D. Harvard 1900.
Emory Leon Chaffee (1885-1975).  Ph.D. Harvard 1911.
Howard Hathaway Aiken (1900-1973).  Ph.D. Harvard 1939.
Anthony Oettinger (b. 1929-03-29).  Ph.D. Harvard 1954.
Sheila Greibach (b. 1939-10-06).  Ph.D. Harvard 1963.
Gerard Michon (b. 1956-03-29).  Ph.D. UCLA 1983.
 
His son  Charles "Santiago" Sanders Peirce  (1839-1914)  made a fleeting mark in  epistemology  by introducing  pragmatism  as a guide for  scientific research  (a viewpoint which is now deprecated).  Twenty years after the hints of  Hermann Grassmann,  Charles S. Pierce published an axiomatization of natural numbers in a paper entitled "On the Logic of Number" (1881).  The modern version  (Peano's axioms)  is usually credited to  Dedekind (1888)  and  Peano (1889).  Peirce based his own  Algebra of Logic  (1885)  on what's now called  Peirce's law.

(2019-01-25)   Enumeration of Finite Rings
How many different rings with  n  elements are there?

To count the number of rings of order  n  (up to isomorphism)  we first  factor  n  into  m  primes :

n   =   q₁^k₁ q₂^k₂ ... q_m^k_m

A ring is unital  (resp. commutative)  if and only if  all  its components are.  So,  the same enumeration method applies either to unrestricted rings or to those which are required to be unital and/or commutative.  The ensuing four possible types of enumerations are tabulated below in separate columns.

The only possible additive group for a ring of  prime  order  p  is the  cyclic group  C_p .  As the  characteristic  of a finite ring must divide its order,  it's either 1 or p.  This yields only two possible rings.  Both are commutative:

• The  ring of square zero   C_p(0)   is not unital:   " x,  " y,   x y  =  0
• The  Galois field    GF(p)  =  F_p  =  /p   is a unital ring,  of course.

There are  11  different rings of order  p²,  for any prime p.  Of those,  9  are commutative and  4  are unital  (all the unital ones are commutative):

• 3  such rings are decomposable:   (C_p(0))²,   C_p(0) × F_p  and  (F_p)²
  Only the last one is unital.  All three are commutative.
• 5  other rings over the additive group   (C_p)²   are indecomposable.  These,  too,  are  algebras  of dimension  2  over the finite field  F_p :
  • The two unital ones are commutative:   F_p²   and   GR(p,2)
  • One is commutative but not unital.
  • The other two are neither commutative nor unital.
• The  3  indecomposable rings over the group  C_p²  are commutative.
  Only   /p²   is unital.

The enumeration for order  p³  is far more delicate.  Preliminary results were published in  1947  by  Robert F. Ballieu,  a leading professor at  Université catholique de Louvain  and a former foreign student at  ENS  (Ulm, 1936).  A first attempt at a complete enumeration was published in  1973  by  Gilmer  &  Mott.  It was corrected by  Antipkin  &  Elizarov  in 1982:  20 such rings are decomposable and either  32  (for p=2)  or  3p+30  (for p≥3)  are not.

The entries in  bold  were first obtained by  Christof Nöbauer  before 2002.

Because the corresponding enumerations are  multiplicative functions  (entirely determined by their values at prime-powers)  the above table is sufficient to enumerate rings of all orders below  64  (except  32  for rings not required to be unital).  This yields the following four tables:

By comparing the last two tables,  we see that the  unique  smallest unital noncommutative ring has  8  elements;  it's the SUN ring studied  above.  Next up,  there are  13  distinct noncommutative unital rings of order  16.  Noncommutative unital rings exist only for orders  divisible by a cube :

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304... (A046099)

Indeed,  if   n  =  k p³ ,  then one example of a noncommutative unital ring of order  n  is given by the  direct product   SUN_p × /k

All unital rings of  cubefree  order  n  (A004709)  are commutative because the above enumerations show that there are as just as many  commutative unital rings  of order  n  as there are  unital rings  of the same order.  (HINT :  That much is true when  n  is a prime or the square of a prime.)

Likewise all rings of  squarefree  order  (A005117)  are commutative.  Noncommutative rings exist only for orders  divisible by a square :

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148... (A013929)

Finite rings   |   Matrix rings
Classification of Finite Rings of Order p²  by  Ben Fine  (MAA 64, 4, 248-252, Oct. 1993).
 
How many unital rings with 35 elements?  (Math Stack Exchange, 2016-12-12).
Finding all rings of order p³   (Math Stack Exchange, 2014-10-05).
Counting rings of order p³   (Math Stack Exchange, 2015-01-19).
Structure theorem for finite rings   (Math Stack Exchange, 2013-04-21).
Rings and modules of finite order   (Math Stack Exchange, 2011-06-09).
Classification of finite commutative rings   (Math Stack Exchange, 2009-11-29).
Enumerating finite local commutative rings effectively  by  Daniel Briggs  (Math Stack Exchange, 2017-08-17).
 
Tables of finite commutative local rings of small orders  by  Andrzej Nowicki  (2018-08-08).
Finite local rings  by  G. Ganske  &  B.R. McDonald  Rocky Mt. J. Math.  3, 4, 521-540  (Fall 1973).
 
Anneaux finis: ...rang 3...  by  Robert F. Ballieu.  Ann. Soc. Sci. Bruxelles, 61, 222-227 (1947).
Finite Associative Rings  by  R. Raghavendran.  Compositio Mathematica, 21, 2, 195-229 (1969).
Associative rings of order p³   by  Robert Gilmer & Joe Mott.  Proc. Japan Acad., 49, 10 (1973).  Superseded.
Rings of order p³  by  V.G. Antipkin  &  V.P. Elizarov.  Sib. Math. J.  23, 4, 457-464  (July 1982).

(2019-01-25)   Simple rings  do not possess any  nontrivial  ideals.
They're called  quasi-simple  by  Bourbaki.

Ideals  are to rings what divisors are to integers or what  normal subgroups are to groups,  inasmuch as they allow a standard decomposition when nontrivial examples exist.  Otherwise,  we indicate irreducibility to anything simpler by calling those things  simple rings,  primes  or  simple groups.

{0}  isn't considered a simple ring,  for the same reason  the number  1  isn't called a prime  or  {e}  isn't considered a simple group:  If this convention wasn't made,  fundamental theorems would be tougher to state  (and use).

Simple ring

(2019-02-12)   Coprime ideals in a  commutative  ring  A:
Two  ideals  I  and  J  are called  coprime  when  I + J = A.

The term  comaximal  is more rarely used for the same concept,  which generalizes  coprime integers  in a way which allows the following generalization of the  Chinese remainder theorem.

Generalized Chinese Remainder Theorem

Coprimality
Ideals and the Chinese Remainder Theorem   Mathematics Stack Exchange  (2018-03-01).

(2019-02-13)   Bezout Rings
Rings in which the sum of two  principal ideals  is again principal.

Bézout domain   |   Bézout's lemma   |   Etienne Bézout (1730-1783)

(2019-01-18)   Primary Ideals
They are to  prime ideals  what prime-powers are to prime numbers.

A  primary decomposition  is an expression of an ideal as an intersection of finitely many primary ideals.  It's analogous to the factorization of a positive integer into a product of prime-powers  (fundamental theorem of arithmetic).

The fundamental theorem about primary decompositions is the  Lasker-Noether theorem,  which was the main motivation for the definition of  Noetherian rings  (introduced  below).

By definition,  an ideal  I  is said to be  primary  when,  for any product x y  in it,  either  x  or some power of  y  is also in  I.

Primary ideal

(2006-04-27)   Cauchy Product
A well-defined internal operation among sequences in a ring.

The Cauchy product of two sequences  (a₀ , a₁ , a₂ , ...)  and  (b₀ , b₁ , ...)  of elements from a ring  A  is the sequence   (c₀ , c₁ , c₂ , ...)   where:

	n
cn   =	å	ai bn-i
	i = 0

Namely:  c₀ = a₀ b₀ ,   c₁ = a₀ b₁ + a₁ b₀ ,   c₂ = a₀ b₂ + a₁ b₁ + a₂ b₀ ,  etc.

The set  A ,  of the sequences whose terms are elements of the ring  A  has the structure of a  ring  (dubbed  formal power series  over A)  if endowed with  direct  addition  (the n-th term of a sum being the sum of the n-th terms of the two addends)  and the  Cauchy multiplication  defined above.

The set  A^{( )}  consists of those sequences which have only  finitely many nonzero terms.  It forms a subring of the above ring, better known as the  [univariate]  polynomials over  A, denoted  A[x]  and discussed next.

(2006-04-06)   A[x] :  Ring of  formal  polynomials over a ring  A
It's endowed with component-wise addition and  Cauchy multiplication.

A  finite  sequence of elements of a ring  A  (or, equivalently, a sequence with finitely many nonzero elements)  is called a  polynomial  over  A.  The set of all such polynomials is a ring  (often denoted  A[x]  where  x  is a "dummy variable")  which is a subring of the aforementioned ring of "formal power series", under direct addition and Cauchy multiplication.

Each term of the sequence defining a polynomial is called a  coefficient.  The  degree  of a polynomial is the highest rank of its nonzero coefficients  (ranks start at zero).  The null polynomial ("zero") has no nonzero coefficients and its degree is defined to be -¥  (negative infinity).  Thus,  for polynomials over an  integral ring  (a ring without divisors of zero)  the degree of a product is always the sum of the degrees of the  factors.

If the ring  A  is commutative,  the polynomials so defined match very well the intuition we may have acquired by using  X  as the symbol for a so-called  unknown variable  and forming new expressions by addition and multiplication using  X  and some  constants  taken from the ring  A.

In the noncommutative case,  this intuition fails unless we imagine that  X  commutes with every element of  A.  This strange thing is what it  means  to state that the product of two polynomials is equal to their  Cauchy product.

Formal Polynomials  vs.  Polynomial Functions :

To a polynomial  (a₀ , a₁ ... a_n )  of degree  n,  we associate a function  f :

	n
f (x)   =	å	ai xi
	i = 0

However, that function and the polynomial which defines it are two different things entirely...  For example, over the finite field  GF(q),  the  distinct  polynomials  x  and  x^q  correspond to the  same  function.

In other words, the map from polynomials to  polynomial functions  need not be injective.  However, that map is indeed injective in the special case of polynomials over  ordinary signed integers  or any superset thereof,  including rational, real, surreal or complex numbers.  (As is the case for  p-adic integers and  p-adic numbers.)

Whenever the distinction between a polynomial and its associated function must be  stressed,  the former may be called a  formal polynomial.

Similarly, infinite sequences of coefficients are called  formal power series  Unlike polynomials, those may or may not be associated with a  convergent  power series  which would define a proper function...

Over a noncommutative ring, the concept of polynomials doesn't break down, but the above association of a polynomial with a function is dubious at best,  in particular because the value of a product of polynomials at some point need not be the product of their values at that point.

Polynomial Functions over a Noncommutative Ring

Arguably,  some polynomial functions are not even associated with a formal polynomial in the above sense.  Indeed,  we could consider functions which are obtained from variables and constants by some fixed recipe involving a finite number of additions and multiplications,  without assuming any type of commutativity among constants or variables.  This topic is under-investigated in the literature.

Twisted Polynomial Ring

It's the set of univariate polynomials with coefficients appearing formally at the left of the powers of the variable  X  if multiplication is defined using a  nontrivial  automorphism  s  in the ring of coefficients  (e.g.,  the Frobenius map  in a commutative ring of coefficients of nonzero  characteristic)  with the following  postulated  formal identity:

X b   =   s(b) X

In other words,  the  n-th coefficient of the  twisted product  is:

	n
cn   =	å	ai  s ( bn-i )
	i = 0

Since  s  isn't trivial,  multiplication of twisted polynomials isn't commutative even when the ring of coefficients is  (it stays  Noetherian). 

Polynomial functions   |   Polynomials   |   Twisted polynomials
Noncommutative division rings with prime characteristics  by  Ryan Schwiebert, Ph.D. 2011  (2015-01-27).

(2022-10-15)   Bombieri Inner Product
Bombieri product of two homogeneous multivariate polynomials.

Bernard Beauzamy (1949-)   |   Jérôme Dégot (thèse, 1995)

(2019-01-18)   Noetherian Rings
Rings which don't contain any infinite  ascending chains  of  ideals.

Four concepts can be defined,  which use the concept of an  ascending chain,  defined below.  They coincide for  commutative  rings:

• Weakly Noetherian :   No infinite ascending chain of  ideals  exists.
• Left-Noetherian :   No infinite ascending chain of  left-ideals
• Right-Noetherian :   No infinite ascending chain of  right-ideals
• Strongly Noetherian :   Both  left-Noetherian  and  right-Noetherian.

A  weakly Noetherian  ring need not be either  left-Noetherian  or  right-Noetherian,  since the lack of a chain of sets which are both left-ideals and right-ideals doesn't disallow a chain of sets with either property.

An  ascending chain  of sets is defined as a sequence of sets  I_n  where every set is contained in its successor,  which is to say:

I₀   Í   I₁   Í   I₂   Í   I₃   Í   ...   I_n   Í   I_n+1     ...

That concept can be defined for any objects endowed with some  partial ordering.  So can the  mirror notion  of a  descending chain.  A chain is dubbed  infinite  if it contains infinitely many  distinct  terms.  Otherwise,  there will be an index  N  after which all terms are equal to each other.  Such a chain is said to be finite,  to terminate or to stabilize.
 
The statement that no infinite ascending  (resp. descending)  chain exists is generically known as the  ascending chain condition  (ACC)  for the relevant ordering  (resp. descending chain condition, DCC).

The non-commutative case is under-investigated because the  Lasker-Noether theorem  (discussed in the next section)  doesn't apply to it.  Emmy Noether  herself gave a non-commutative example of a  right-Noetherian  ring where some ideals don't have a  primary decomposition.

Equivalently,  Noetherian rings can be defined as:

• Rings in which every set of ideals has a  maximal  element.  (with respect to set-inclusion only;  it needn't be a  proper  maximal ideal).
• Finitely-generated rings.  (In the commutative case,  at least.)

Trivially,  all  finite  rings are Noetherian.  Not all  Bézout rings  are.

Another example of a (commutative) Noetherian ring is the ring of integers  .  Indeed,  all its ideals are of the form  n  (the  principal ideal  generated by  n).  Thus,  any ascending chain of ideals corresponds to a decreasing sequence of divisors,  which is necessarily finite,  so the  ACC  holds.

More generally,  any  principal ring  (or PID)  is Noetherian.  In particular,  every  field  is a  Noetherian ring.

Hilbert's Basis Theorem  (Hilbert, 1888) :

Univariate polynomials  over a  Noetherian ring  form a  Noetherian ring.
Corollary  (by induction) :   So do  multivariate  polynomials.

This theorem predates the formal introduction of Noetherian rings by about thirty years!  David Hilbert (1862-1943) gave a nonconstructive proof  (1888)  in the special case of univariate polynomials over a field  (fields are Noetherian)  which states the amazing fact that the ring of polynomials over a field is finitely generated.  I am led to believe that Hilbert used only Noetherian concepts in his proof,  so the modernized theorem remained recognizable and retained its original name  (although the corollary is trivial only when the result is stated in Noetherian terms).

The theorem is true also in the noncommutative case.  Let's give a proof which doesn't depend on commutativity:

Let  A  be a Noetherian ring  (using one of the  four possible definitions  presented above).  A[X]  is the ring formed by all  univariate polynomials  with coefficients in  A.  Consider any  ascending chain  of ideals in  A[X] :

I₀   Í   I₁   Í   I₂   Í   I₃   Í   ...   I_n   Í   I_n+1     ...

To prove the theorem,  we must establish that this chain cannot be infinite  (i.e., it must stabilize after some large enough index).

Well,  we may define an ideal  J_ij  in  A  as the set of all the coefficients of  X^j  in the polynomials of  I_i  with  degree  i  or less.  It's not difficult to prove that every  J  is an ideal  (one-sided or double-sided)  in the same sense as every  I  is.  (That exercise is left to the reader.)

Furthermore,  it's clear that,  if  i ≤ i'  and  j ≤ j'   then  J_i,j   Í   J_i',j'

In spite of its modern terminology,  the above proof is probably quite close to the nonconstructive proof originally proposed by Hilbert,  who was motivated by the abstract generalization of results in the  theory of invariants  which had been painstakingly derived by  Paul Gordan (1837-1912)  of Clebsch-Gordan_coefficients fame.  Legend has it that Gordan paid a strange compliment to this beautiful proof:

This is not mathematics,  this is theology.

Gröbner bases   (Bruno Buchberger,  1965)

Noetherian ring   |   Primary ideal   |   Emmy Noether (1882-1935)
Hilbert's basis theorem (1888)   |   Proof on AoPS   |   David Hilbert (1862-1943)
Gröbner bases (Buchberger, 1965)
Examples of Noetherian rings  in  Commutative Algebras  lectures by  Simon J. Wadsley.
Noetherian rings are the finitely-generated rings  by  Tai-Danae Bradley  (Math3ma, 2015-05-12).

(2019-01-21)   Artinian rings are Noetherian  (but the converse isn't true).
Rings which don't contain any infinite  descending chains  of  ideals.

For rings,  surprisingly,  the Artinian  descending chain condition  (DCC)  implies the Noetherian  ascending chain condition  (ACC)  but turns out to be  strictly stronger.  The following statement clarifies the situation and makes the independent study of Artinian rings all but superfluous:

An Artinian rings is a Noetherian ring
where every non-invertible element is  nilpotent.

Artinian rings may also be characterized as Noetherian rings whose  Krull dimensions  are not positive.

The importance of Artinian rings stems from the fact that it's the natural concept to use in the  Artin-Wedderburn classification theorem.

An Artinian ring is a finite product of local rings  (Chriestenson, Tuley, Wane).

Artinian ring   |   Artinian rings and modules  (AoPS = Art of Problem Solving)
Artin-Wedderburn theorem   |   Emil Artin (1898-1962)

(2019-01-18)   Lasker-Noether Theorem   (Lasker 1905,  Noether 1921)
Every  commutative  Noetherian ring  is a  Lasker ring  (Laskerian ring).

By definition,  a  Lasker ring  is a commutative ring in which any  ideal  has a  primary decomposition  (which is to say that it's the intersection of finitely many  primary ideals).  In other words,  a generalization of the  fundamental theorem of arithmetic  holds in such rings.  In 1905,  Emanuel Lasker  considered only  polynomials  and convergent power series over a  field.  He proved that the rings they form have that property.  In 1921,  Emmy Noether  generalized the theorem to all  commutative Noetherian rings.

I don't know any non-Noetherian Lasker ring,  but that's not ruled out.

Lasker didn't point out the unicity of the decomposition,  which was established in 1915 by  Francis Macaulay (1862-1937).

The algorithm commonly used to perform actual  primary decompositions  is due to  Grete Hermann (1901-1984).  It was part of the doctoral work she completed under  Emmy Noether  at  Göttingen  in 1926.

Primary decomposition
 
"Zur Theorie der Module und Ideale" by Emanuel Lasker, Math. Ann. , 60 (1905) pp. 19-116
"Idealtheorie in Ringbereiche" by Emmy Noether,  Math. Ann. , 83 (1921) pp. 24-66

Emanuel Lasker (1868-1941,  Ph.D. 1902)  was a talented German mathematician,  mostly remembered as the longest-reigning Chess world champion  of the modern era.  He held the title officially for 27 years (1894-1921)  including two periods  (1898-1906 & 1911-1920)  when he wasn't challenged,  either because of world events  (WWI)  or because nobody could raise the money to do so.  Following in the footsteps of his older brother Berthold, Emanuel Lasker depended on game-playing for financial support from an early age  (mostly chess but also card games;  he was a world-class bridge player).  Lasker was one of the strongest  Go  players outside of Asia.  He had been introduced to Go by his close friend and distant relative  (third cousin twice removed)  Edward Lasker (1885-1981)  the amateur chess champion who co-founded the  American Go Association (AGA)  in 1935.
 
Lasker tried to retire from competitive chess and exhibitions in the late 1920s but had to return to it once everything he owned was confiscated by the Nazi regime.  He was fairly successful in spite of his age but his health degraded and he died in dire straits in the US.
 
Lasker's 1905 paper on ring theory marks the high-point of his mathematical career and represents his only lasting mathematical contribution.  It's  David Hilbert (1862-1943)  who had advised him to undertake doctoral studies (1900) at  Erlangen  under  Max Noether (1844-1921)  the father of Emmy Noether (1882-1935).  Max Noether was one of the early leaders in  algebraic geometry,  at the heart of which Lasker's theorem can be construed to be  (with or without the generalization due to Emmy Noether).  Especially if we recast it in the following form:  Any  algebraic variety  is the union of finitely many  irreducible components.
 
Lasker became a close friend of  Albert Einstein (1879-1955)  who esteemed him highly.  Einstein lamented that Lasker's great talents had been wasted on Chess.

(2019-01-22)   GCD Domains
In them,  two elements always have a  greatest common divisor  (GCD).

As the name implies,  a  GCD domain  is a commutative ring.

GCD domain

(2019-01-22)   Unique Factorization Domains   (UFD)
GCD domains  with no  infinite ascending chains  of  principal ideals.

Bourbaki  called such a ring a  factorial ring,  which is now the only term used in French  (namely,  anneau factoriel ).

In a ring,  two related concept  (irreducible and prime)  can be defined which coincide only in the case of a  factorial rings  (UFD).  In such a structure,  a straight counterpart of the  fundamental theorem of arithmetic  hold,  which justifies the term of  unique factorization domain.

• An  irreducible  element  q  is an element which only has  trivial  factorizations  (i.e.,  it can only be written as a product of two factors if at least one of those is a  unit).
• A  prime  element  p  is an element which can't divide a product of two factors unless is divides one of them.

The ring  [Ö5]  (pronounced    adjoined  root  5)  isn't a  UFD  because it contains  irreducible  elements which aren't  prime.  For example,  the integer  2  divides the following product without dividing either factor:

4   =   (-1 + Ö5) (1 + Ö5)

In 1847,  the failure to distinguish between those two concepts famously led the French physicist  Gabriel Lamé (1795-1870; X1814)  to propose an erroneous proof of  Fermat's last theorem.  A few years earlier  (1843)  Eduard Kummer (1810-1895)  had already identified this trap,  which would become one of the key motivations for the development of  ring theory.

Unique factorization domain  (factorial ring)

(2019-04-23)   Dedekind Domain   (or  Dedekind Ring)
Integral domain where every nontrivial ideal is a product of prime ideals.

An integral domain is a Dedekind domain if, and only if, every nonzero fractional ideal is invertible.

Dedekind domain   |   Fractional ideal   |   Class group

(2006-04-05)   GR(q,r) :  Galois ring of characteristic  q = p^m and rank  r
The modulo-q polynomials modulo an irreducible polynomial modulo p.

Let  q  be a power of a prime  p.  Let   f  be some  monic  polynomial modulo q, of degree  r,  which is  irreducible modulo p  (i.e.,  f (x) mod p  never vanishes).  The  Galois ring  of characteristic  q  and rank  r  is defined as:

GR(q,r)   =   (/q)[x] / f (x)

Finite noncommutative rings can be described as algebra over Galois rings.

Encyclopedia of Mathematics   |   ScienceDirect
 
About Galois rings   (Math Stack Exchange, 2013-03-24).

(2014-12-06)   Hilbert rings   =   Jacobson rings
In those, every prime ideal is an intersection of  maximal ideals.

In particular, an Hilbert subring ring is the intersection of all maximal ideals containing it.

They were called  Hilbert rings  by Serre  and/or  Bourbaki  because of the connection with Hilbert's  Nullstellensatz.  Outside the Bourbaki sphere of influence, the attribution fo Jacobson seems to be preferred.  The qualifier  maximal  is sometimes replaced by  primitive

Jacobson ring   |   Hilbert's Nullstellensatz

(2019-05-03)   Abelian Rings
Rings in which every  idempotent element  is  central.

Abelian rings need not be commutative but they have some properties of commutative rings.

Why aren't commutative rings called abelian?  (Math Stack Exchange, 2014-06-04).

(2018-01-25)   Local rings   (Krull, 1938)
Local algebra  is the study of commutative local rings and their modules.

A  local ring  is a ring which has only  one  maximal ideal.

This concept was introduced in 1938 by  Wolfgang Krull (1899-1971)  who called them  Stellenringe  in German.  The English term  local ring  was coined by  Oscar Zariski (1899-1986).

Local rings (1938)   |   Wolfgang Krull (1899-1971,  Ph.D. 1922)
Localization of a ring   |   Completion
 
Cohen-Macaulay rings (Macaulay 1916, Cohen 1946)
Francis Macaulay (1862-1937)   |   Irvin Cohen (1917-1955)
 
Noncommutative local ring (Mathematics Srack Exchange, 2013-11-17).

(2019-02-03)   Embeddings

Ore condition   |   Øystein Ore (1899-1968, Ph.D. 1924)

(2020-04-30)   Topological Ring
When the  topology  makes addition and multiplication  continuous.

When a ring  A  is a topological space,  its  cartesian square  A×A  is considered endowed with the  product topology  (of Tychonoff).  The addition and multiplication are functions from  A×A  to  A.  When both are continuous,  A  is said to be a  topological ring.

Topological ring

(2020-05-09)   Involutive Ring  (*-ring)
What a  ring  is called when endowed with an involution which is  both  an additive automorphism and a multiplicative anti-automorphism.

By definition,  an  involution  is just a  bijection  equal to its own inverse.  In the context of  *-rings,  the image by the aforementioned involution of an element  x  is denoted  x*  (pronounced x-star)  and is called the  adjoint  of  x  (or its  conjugate).  It's now customary to call  the  involution the function which maps an element to its adjoint  (I still call it  conjugation  at times).

In elementary texts,  the word  transconjuguate  is also used as a reminder that the adjoint of a  matrix  of  complex numbers  is the transpose of the matrix formed by the complex conjugates of its elements.  There's little need to do transposition without conjuguation  (or vice-versa).  It's a bad idea to use a postfixed star to denote conjugation without transposition,  which would call for another postfix symbol to denote the adjoint  (many physicists use a dagger).  I find it convenient to use the star to denote all adjoints  (including complex conjugates).  Especially if  hypercomplex numbers  are involved.

That's to say that the following axioms are postulated:

• X**   =   X     (i.e.,  "conjugation" is an  involution).
• (X + Y)*   =   X* + Y*     (additive homomorphism).
• (X Y)*   =   Y* X*     (multiplicative antihomomorphism).

*-Rings

(2020-04-30)   Etale homomorphisms of rings

Etale morphism   |   Formally étale homomorphisms of rings